[FOM] Mathematics and formalizability

[Martin Davis]

At 12:21 PM 7/30/2004, Vladik Kreinovich wrote:
>There are two different notions of what mathematics is.
>
>We mathematicians usually define mathematics by the level of rigor, while
>others define mathematics by objects of study: if it is about abstract
>mathematical objects, it is mathematics, eevn if there is no rigor.
>
>Mathematicians all (99% probably) agree that mathematics is something that is
>formal or at least formalizable.

This narrow approach flies in the face of history and mathematical 
practice. Most mathematicians aim for "rigor" but give little if any 
thought to formal systems. Rigor (and ultimately formalizability) can be 
regarded as goals, but that's all. And as E.T. Bell once remarked 
"Sufficient unto the day is the rigor thereof".

Here are some examples of mathematical activity that don't meet Vladik's 
requirements;

1. Euler sums 1/n^2 to pi^2/6 by a heuristic argument factoring the power 
series for sine as though it were a polynomial and using the relation 
between symmetric functions of the roots of a polynomial and its  coefficients.

2. Italian algebraic geometry

3. Heaviside operator calculus

Martin